On concentration of real solutions for fractional Helmholtz equation

This paper studies the nonlinear fractional Helmholtz equation \begin{equation}\label{main} (-Δ)^{s} u-k^{2} u=Q(x)|u|^{p-2}u, $\mathrm{i}\mathrm{n}$$\mathbb{R}$^{N},~~N\geq3, \end{equation} where $\frac{N}{N+1}<s<\frac{N}{2}$, $\frac{2(N+1)}{N-1}<p<\frac{2N}{N-2s}$ are two real exponents, and the coefficient $Q$ is bounded continuous, nonnegative and satisfies the condition \begin{equation} \mathop{$\mathrm{l}\mathrm{i}\mathrm{m}\mathrm{s}\mathrm{u}\mathrm{p}$}\limits_{|x|\longrightarrow\infty}Q(x) <\mathop{$\mathrm{s}\mathrm{u}\mathrm{p}$}\limits_{x\in$\mathbb{R}$^{N}}Q(x). \end{equation} For $k>0$ large, the existence of real-valued solutions for (\ref{main}) are proved, and in the limit $k⟶\mathrm{\infty }$, sequence of solutions associated with ground states of a dual equation are shown to concentrate, after rescaling, at global maximum points of the function $Q$.